VMO 1996 — Division A Question 6

Let $x,y,z,w$ be nonnegative real numbers satisfying
$$2(wx+wy+wz+xy+xz+yz)+wxy+xyz+yzw+zwx=16.$$
Prove that
$$3(w+x+y+z)\ge2(wx+wy+wz+xy+xz+yz).$$